Pythagoras' Theorem

• we are learning to find a missing side length in a right-angled triangle when we know the other two side lengths.

• This is useful because there are times when we might know that a particular triangle is right-angled and know two of the lengths, but need to know the third length.
• Also, because the rule we will discover only works when it is a right-angled triangle, we can use it to check whether a particular triangle has a right-angle if we know all three side lengths.

• A functional (real-life) application is in navigation: we can work out how far we have gone 'as the crow flies' (on a direct route) if we know how far North and East we have travelled.
• Another functional application is to design shelf brackets or tell how far a ladder will safely reach.
• Some builders use this rule to check whether a door frame is exactly 'true' meaning that it has a perfect right-angle in the corner. It's more accurate than using a protractor to measure the angle.
• You could use the same trick to check a picture frame is true.
• TV and computer screens are sold by describing the diagonal length across the screen. Knowing this length and the 'aspect ratio', we could find out exactly how tall and wide the screen is.
• Engineers and scientists use this rule to find the result of two or more forces acting on an object.

• This skill leads to a topic called trigonometry (literally 'triangle measurement') which allows us to find out other angles and lengths in triangles.
• The rule about lengths is used in a wide range of other mathematics including coordinates and lines, vectors, complex numbers, ...

• A 'free gift' with this skill is that we can now also find the length of the diagonal in a cuboid.
• Another 'free gift' is that we can find the distance between two points given their Cartesian coordinates.

• This could help you if you want to work in building, design, engineering, science, mathematics...

• Are we ready?
  • Can you write down the first 10 square numbers without having to work them out?
  • Can you make a list of the next ten square numbers from 11x11 to 20x20 ?
  • What is ?
  • Can you draw triangles accurately?
  • Can you measure lengths and angles accurately?
  • Can you substitute into equations and formulae?
• Let's be sure you can...
• Before you start you need to know:
• You will have a deeper understanding if you also know:

• Can you figure it out yourself from these examples?
• Try to predict the next one aloud or on a mini-whiteboard.

• Here are some examples of people getting it right:
• Here are some examples of people getting it wrong in typical ways:
• Here are some more examples. Did they get it right or wrong? Explain how you know!

• What would this one be? Tell your learning partner. Convince them you're right.
• Explain how you know.
• How would you explain this to someone who was new to it?

• A really good introduction to Pythagoras' Theorem at Skoool.co.za
• One way to do this is...
• Another approach might be...
• A useful shortcut is to...
• This works because...
• It doesn't work when...
• An exception is...
• Watch out for...
• A common mistake is...
• You can check your result by...
• We can prove this works by any one of hundreds of methods.
  • Alexander Bogomolny shows nearly a hundred proofs (some with some lovely animated applets) at the Pythagorean Theorem page of cut-the-knot.org
  • Euclid's proof illustrated by an applet by Jim Morey is also hosted on cut-the-knot.org
  • The easiest proof for pupils to understand is probably the two big squares proof, also known as the tangram-type proof. A useful demonstration of the Pythagorgrams proof here.
  • It can be done using a piece of 14cm square card. A
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• Some straightforward examples.
• Some harder examples.
• Some mixed examples.
• Some non-examples to spot and some mixed questions with redundant, insufficient or contradictory data.

• Check you've mastered this skill by...
• Show you understand by explaining...
• Prove you're an expert in... by...
• Returning to the Learning Objective(s),...

• Next we could learn...
• This leads to...
• Now try...

Unsorted links

• NSDL Pythagoras resources

Classroom resources:

Pythagoras' Theorem.jnt

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